2.3 Notes – Real Zeros of Rational Polynomials/ 2.5 Fundamental Theorem
Remember To divide polynomials:
1. Write the dividend and the divisor in descending powers of the variable.
2. Insert zeros as placeholders for missing powers.
*Types of polynomial division Long division: can be used with any divisor
Synthetic division: can only be used if the divisor is in the form x  c
EX1a:
Fundamental Theorem of Algebra: for any polynomial function, the degree of the
polynomial equals the number of zeros.
For example:
has ____ zeros
has ____ zeros
has ____ zeros
Remainder Theorem: If a polynomial f (x) is divided by x  k , the remainder
is r  f (k )
EX2: f ( x)  3x 3  8 x 2  5 x  7 let x  2
-2
3 8 5 -7
-6 -4 -2
3 2 1 -9 (remainder)
f (2)  24  32  10  7  9
f (2)  9
Factor Theorem: A polynomial f (x) has a factor ( x  c) if and only if f (c)  0 .
EX3: f ( x)  2 x 4  7 x 3  4 x 2  27 x  18 ; Show that ( x  2) and ( x  3) are factors.
2
2 7 -4 -27 -18
4 22 36 18
2 11 18 9
0 (remainder) r = 0, so x  2 is a factor
-3
2
2
11 18 9
-6 -15 -9
5
3 0 (remainder) r = 0, so x  3 is a factor
1. Remainder r gives the value of f at x = k. [r  f (k )]
2. If r = 0, ( x  k ) is a factor of f (x).
3. If r = 0, (k, 0) is an x-intercept of the graph of f.
1
The Rational Zero Test: relates the possible rational zeros of a polynomial to the leading
coefficient and to the constant term of the polynomial. If there are any irrational or
imaginary zeros, you will not find them using the Rational Zero Test – hence the name.
Possible rational roots =
factors of the constant term
factors of the leading coefficient
1. Use trial and error to determine which, if any, are actual zeros.
2. Once the partial quotient is a quadratic, either factor or use the quadratic formula
to find remaining zeros
EX 4a: Find the rational zeros of f ( x)  x 3  x  1
EX 4b: f ( x)  2 x 3  3x 2  8 x  3
EX 4c: f ( x)  2 x3  x 2  x  4
*Other test for finding zeros:
DesCartes Rule of Signs:
1. The number of positive real zeros of f is either equal to the number
of variations in sign of f (x) or less than that number by an even integer.
2. The number of negative real zeros of f is either equal to the number
of variations in sign of f ( x) or less than that by an even integer.
Variation in sign: 2 consecutive coefficients have opposite signs (+,-)
The number of imaginary zeros + positive zeros + negative zeros will always equal the
degree of the polynomial.
EX 5a: f ( x)  3x 3  5 x 2  6 x  4
5b:
5c:
2
Finding ALL roots:
1. Use DesCartes Rule of Signs to determine the number of possible real roots.
2. Use the Rational Zero Test to determine any rational roots that may exist.
3. Divide the original polynomial by factors associated with the known possible
rational roots.
4. Factor further (or use the quadratic formula) to find remaining roots of function.
EX 6a:
EX 6b:
EX 6c:
Note that imaginary roots come in conjugate pairs (think about quadratic formula…)
EX 7: Find all zeros of
, given
is a root.
Then write your answer as linear factors.
3
Given roots, write the polynomial function.
(hint…go back to factor form (x – root), then multiply)
EX 8a: 1, -2, 5
8b:
8c:
8d:
4